## Overview

In addition to what has been reviewed in meeting structural requirements, analyses of product shapes also includes factors such as the size available equipment can handle including thicknesses and product complexity and capability to package and ship to the customer

(Chapters 1 and 2). The ability to achieve specific shapes and design details is dependent on the way the process operates and plastics to be processed. Generally the lower the process pressure, the larger the product that can be produced. With most labor-intensive fabricating methods, such as RP hand lay-up with TS plastic there is virtually no limit on size.

An important requirement concerns meeting dimensional tolerances of shaped products. Reported are different shrinkages for different plastics per standard tests that may have a relation to the designed product. The probability is that experience with prototyping will only provide the true shrinkage conditions of the shaped products. Minimum shrink values are included in the design of mold cavities and die openings so that if the processed plastic does not meet required dimensions all that is required is to cut the metal in the tools.

If the reverse occurs, expensive tool modifications may be required, if not replacing the complete tool. It is vital to set up a complete checklist of product requirements, to preclude the possibility that a critical requirement may be overlooked initially. Fortunately there are occasions where changes in process control during fabrication can be used to produce the required dimensional product.

### Filament Wound Shape

Filament winding (FW) shapes are principally circular (cylinders, pipes, tubing, etc.) or enclosed vessel (storage tanks, oxygen tanks, etc.). They produce spherical, conical, and geodesic shapes. The fabricating process permits tighdy controlled fiber netting orientation and exceptional quality control in different fiber-resin matrix ratios required by design. Structures can be fabricated into shapes such as rectangular or square beams or boxes, longitudinal leaf or coil springs, etc. Filaments can be set up in a part to meet different design stresses.

There are two basic patterns used by industry to produce FW structures, namely, circumferential winding and helical winding. Each winding pattern can be used alone or in various combinations in order to meet different structural requirements. The circumferential winding pattern involves the circumferential winding at about a 90° angle with the axis of rotation interspersed with longitudinal reinforcements. Maximum strength is obtainable in the hoop direction. This type of pattern generally does not permit winding of slopes over 20° when using a wet winding reinforcement or 30° when using a dry winding process. It also does not result in the most efficient structure when end closures are required. With end closures and/or steep slopes, a combination of helical and circumferential winding is used.

With helical winding, the reinforcements are applied at any angle from 25° to 85° to the axis of rotadon. No longitudinal filament need be applied because low-winding angles provide the desired longitudinal strength as well as the hoop strength. By varying the angle of winding, many different rados of hoop to longitudinal strengths can be obtained.

Two different techniques of applying the reinforcements in helical windings are used by industry. One technique is the applicadon of only one complete revolution around the mandrel from end to end. The other technique involves a muld-circuit winding procedure that permits a greater degree of flexibility of wrapping and length of cylinder.

### Netting Analysis

Continuous reinforced filaments should be used to develop an efficient high-strength to low-weight FW structure. Structural properties are derived primarily from the arrangement of continuous reinforcements in a netting analysis system in which the forces, owing to internal pressure, are resisted only by pure tension in the filaments (applicable to internal-pressure systems).

There is the closed-end cylinder structure that provides for balanced netting of reinforcements. Although the cylinder and the ends require two distincdy different netting systems, they may be integrally fabricated. The structure consists of a system of low helix angle windings carrying the longitudinal forces in the cylinder shell and forming integral end closures which retain their own polar fittings. Circular windings are also applied to this cylindrical portion of the vessel, yielding a balanced netting system. Such a netting arrangement is said to be balanced when the membrane generated contains the appropriate combination of filament orientations to balance exactly the combination of loadings imposed.

The girth load of the cylindrical shell is generally two times the axial load. The helical system is so designed that its longitudinal strength is exacdy equal to the pressure requirement. Such a low-angle helical system has a limited girth strength. The circular windings are required in order to carry the balance of the girth load.

The end dome design contains no circular windings since the profile is designed to accommodate the netting system generated by the terminal windings of the helical pattern. It is termed an ovaloid: that is, it is the surface of revolution whose geometry is such that fiber stress is uniform throughout and there is no secondary bending when the entire internal pressure is resisted by the netting system.

There is the ovaloid netting system that is the natural result of the reversal of helical windings over the end of the vessel. The windings become thicker as they converge near the polar fittings. In order to resist internal pressure by constant filament tension only, the radius of curvature must increase in this region. It can also be equal to one half the cylinder radius when the helix angle a = 0°, and equal to the cylinder radius when a = 45°. The profile will also be affected by the presence of an external axial force.

In the application of bidirectional patterns, the end domes can be formed by fibers that are laid down in polar winding patterns. The best geometrical shape of the dome is an oblated hemispheroid. Theoretically, the allowable stress level in the two perpendicular directions should be identical. However, the efficiency of the longitudinal fibers is less than that of the circumferential fibers. It is possible to estimate an optimum or length-to-diameter ratio of a cylindrical ease for a given volume.

The filament-wound sphere design structure provides another example of a balanced netting analysis system. It is simpler in some respects than the closed-end cylinder. The sphere must be constructed by winding large circles omni directionally and by uniform distribution over the surface of the sphere. In practice, distribution is limited so that a small polar zone is left open to accommodate a connecting fitting.

The netting pattern required generates a membrane in which the strength is uniform in all directions. The simplest form of such a membrane would have its structural fibers running in one direction and the other half at right angles to this pattern. This layup results in the strength of the spherical membrane being one-half of the strength of a consolidated parallel fiber system.

The oblated spheroid design structure relates to special spherical shapes. Practical design parameters have shown that the sphere is the best geometric shape when compared to a cylinder for obtaining the most efficient strength-to-weight pressure vessel. The fiber RPs is the best basic constituents. Certain modifications of the spherical shape can improve the efficiency of the vessel. One modification involves designing the winding pattern of the fibers so that unidirectional loading can be maintained. In this type of structure, it is generally assumed that the fibers are under equal tension. This type of structure is identified as an isotensoid. The geometry of this modified sphere is called oblated spheroid, ovaloid, or ellipsoid.

The term isotensoid identifies a pressure vessel consisting entirely of filaments that are loaded to identical stress levels. The head shape of an isotensoid is given by an elliptical integral, which can most readily be solved by a computer. Its only parameter is the ratio of central opening to vessel diameter. This ratio determines the variation of the angle of winding for the pressure vessel. During pressurization the vessel is under uniform strain; consequendy, no bending stresses or discontinuity stresses are induced.

A short polar axis and a larger perpendicular equatorial diameter characterize the vessel. The fibers are oriented in the general direction of a polar axis. Their angle with this axis depends on the size of the pole openings (end closures). For glass fiber TS polyester RP vessels levels of 200,000 psi (1.4 GPa) can be obtained.

The toroidal design structure is a pressure vessel made with two sets of filaments symmetrically arranged with respect to the meridians. They meet two basic requirements: (1) static equilibrium at each point, which determines the angle between the two filaments, and (2) stability of the filaments on the surface, which requires the filaments to follow geodesic paths on the surface. When the equation of the surface is given, these two requirements are generally incompatible. One way to reconcile the correct angularity of the filaments (equilibrium) with the correct paths of the filaments (stability) is to take some freedom in determining the geometry of the surface.

Standard engineering analysis can be used. Consider a cylinder of inside radius r, outside radius R, and length L containing a fluid under pressure p. The circumferential or hoopwise load in the wall (t =

thickness) is proportional to the pressure times radius = pr, and the hoop stress:

fh = hoopwise load/cross sectional area = pr/tor = pd/2t (4-24) similarly, the longitudinal stress:

assuming n (R2 - r2) = 27nrt for a thin-walled tube.

Thus this condition of the hoop stress being twice the longitudinal stress is normal for a cylinder under internal pressure forces only. The load in pounds acts on the tube at a distance from one end and a bending moment M is introduced. This produces a bending stress in the wall of the cylinder of:

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